Suppose A = {-4,-3,–2,–1,0,1,2,3,4} and R is defined on A by aRb a² - b² is divisible by 4. Prove that R is equivalence relation on A. Find all distinct equivalence classes of A with respect to R.

Q: (3) Let S be the equivalence relation on (0, 1, 2, 3} x {0, 1, 2} defined by (a, b)S(c, d) if and…

A: S=0,1,2,3×0,1,2=0,0;0,1;0,21.0;1,1;1,22.0;2,1;2,23.0;3,1;3,2

Q: 3. Let the relation R on X = {1,3,5, 6, 8, 9, 11} be such that xRy means "x – y is a multiple of 3".…

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Q: , In each item below, verify that - is an equivalence relation on the given set. (b) Let - be a…

A:

Q: Let R be a relation on the set N of positive integers defined by aRb if and only if the product axb…

A: This question is about equivalence  class .

Q: Let m e Z+. Show that =m is an equivalence relation on Z. Prove that for any integer a,

A: Equivalence relation

Q: Consider the relation K on N defined by Knm iff n <m.Which statement below is true? K is an…

A: It is Irreflexive.

Q: In each item below, verify that ~ is an equivalence relation on the given set. Let S = Z × N. Define…

A: Since you have asked multiple questions, we will solve the Question Q (a) and (b) for you. If…

Q: B if there exists o E Sn so that oao- = B. Prove that For a, B E Sn, let a ~ equivalence relation…

A:

Q: Let S be the equivalence relation on {− 1, 0, 1} × {0, 1, 2, 3} defined by (a, b )S(c, d) if and…

A: Let A  = {-1,0,1}×{0,1,2,3}  i.e A=…

Q: Consider a relation R on Z- {0} defined by the rule that (x, y) E R if and only if xy > 0. a) Prove…

A: Given the relation R on ℤ-0 defined by         x,y∈R if and only if xy&gt;0.

Q: 8) Let A = {2,3,4, 5}, R is defined on A such that (aRB) if and only if (a2 +B+a) is an even number.…

A: Given A={2,3,4,5} We define a relation R on A by (αRβ) if and only if (α2+β+a) is an even number…

Q: Define R on Z+ by aRb if = 5 for some integer k. (a) Prove R is an equivalence relation. (b) What is…

A: The solution is as follows:

Q: 9.24. Let R be an equivalence relation on A = {a, b, c, d, e, f,g} such that a Rc, c Rd, d Rg and b…

A:

Q: Let Z be the set of integers. Consider R = {(x, y) e Z x Z: 7 divides x-y} S Z x Z. Show that R is…

A:

Q: Show that the relation R in the set A of all the books in a library of a college,given by R = {(x,…

A:

Q: Consider the equivalence relation ~ on R given by a ~ b if and only if [a] = [b]. You do not need to…

A: Given that the equivalence relation ~ on ℝ defined as a~b if and only if a=b. Define a function…

Q: 2. Define the relation on S = Z by a~b + a = b mod 5. (1) Prove that is an equivalence relation. (2)…

A:

Q: Prove that the following relations are equivalence relation: On Z× (Z−{0}), with (a,b)∼(c,d) if…

A:

Q: 10. (A is the relation defined on Z as follows: for all x, y = Z, x Ay ⇒x=y (mod 3). Describe the…

A:

Q: Given the relation R= {(1,1), (2.2), (3.3)(4.4), (1.2). (2.1) (3.4). (4.3)) on the set X- (1,2.3.4).…

A:

Q: Let S' = {x € R² : ||x|| = 1} C X. and endow S' with the same equivalence (b) relation - as above.…

A:

Q: Let H be the set of all human beings. Consider the relation R defined on H by xRy if and only if x…

A: R defined on H by xRy if and only if x and y have the same biological mother. reflexivity : x &amp;…

Q: 1. Consider the relation R on the set A = {0,1, 2, 3, 4}, defined by: %3D aRb + a = bc and b= ad,…

A: The given set is A=0,1,2,3,4 and the relation is defined as aRb⇔a=bc and b=ad (a) Check whether R an…

Q: Show that each of the following relations equivalence classes. on Z is an equivalence relation, and…

A: In the given question we have to find the equivalence classes.

Q: 6. Consider the relation R on Z defined by xRy iff x – y= 4n for some n Ɛ Z. (a) Show that R is an…

A:

Q: Let A be a set and let ∼ be an equivalence relation on A. Let x, y ∈ A. Prove that [x] = [y] if and…

A:

Q: 9. E is the binary relation defined on Z as follows: For all m, n E Z, m Enm-n is even. Is this an…

A: Equivalence relation

Q: Define a relation R on the set of all integers Z as follows: for all integers m and n, m R n → m = n…

A:

Q: Let A = {2,4,6,8,10}. The distinct equivalence classes resulting from an equivalence relation R on A…

A:

Q: Let ~ be defined on the set X = {0,1,2,3,4,5,6,7,8,9} by x ~ y + x2 = y? mod 5. Prove that is an…

A: Equivalence relation are those which are reflexive symmetric and transitive

Q: Let S = {0,±1, ±2, ±3,..., 8} and consider the relation R on S defined by (a, b) E R if and only if…

A: 1.) Reflexive : As [ a ]4 = [ a ]4 for all a in S so ( a, a ) lies in R . 2.) Symmetric : If ( a, b…

Q: Let R be an equivalence relation on A - {a,b,c,d) such that a Rc and bRd How many distinct…

A:

Q: ) Let S = {(x, y) E R × R: y – x is an integer} a) Prove the S is an equivalence relation on R b)…

A: Given, S=x,y∈R×R: y-x is an integer

Q: . The relation R on Z defined by a Rb if a? = b² (mod 4) is known to be an equivalence relation.…

A: Consider the given information.

Q: (3) Let S be the equivalence relation on {0, 1, 2,3} × {0, 1, 2} defined by (a, b)S(c, d) if and…

A:

Q: Show that the set T := {(x, y) E Z × Z | x+y is even number } is an equivalence relation on the set…

A:

Q: Let A = {1, 2, 3,4, 5, 6} and let R be an equivalence relation on A.  Suppose that 1R2, 3R5 and 6R3.…

A: Given that A = {1, 2, 3, 4, 5, 6} R as equivalence relation on A Given conditions: 1 R 2, 3 R 5, 6 R…

Q: 4) Suppose R ={(a,b) €Z×Z:b-a is divisible by 3}, show that R is an equivalence relation on Z.…

A:

Q: Consider the relation on Z×(Z\ {0}) defined by (m,n)R(m’,n’) provided that mn’ = m’n. Prove that…

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Q: Select the following relations on Z that are equivalence relations. ○{(a,b) ||a| = |b|} {(a,b) a²b²…

A:

Q: Let A = {-5, -4, −2, 0, 3, 6, 8), and define an equivalence relation R on A as follows: (x, y) E R…

A: Let A = {-5, -4, -2, 0, 3, 6, 8}, and define an equivalence relation R on A as follows: (x, y) in  R…

Q: Let z be the set of integers and R be the equivalence relation on ZxZ defined by: (a.b)R(c.d) if and…

A: Given that R is a relation on Z×Z defined by (a,b) R(c,d) if and only if a+d=b+c.

Q: (b) By the Equivalence Relation Theorem, there is an equivalence relation R on Z50 such that P is…

A: A relation from X to X is called a relation in X.  If R is in X and: xRx ∀x ∈ X ⇒ R is refelexive…

Q: 14. For a,b E Z, define a ~ b if and only if 4a + b is a multiple of 5. Show that defines an…

A:

Q: (c) Let ~ be a relation defined on Z by a R b if and only if a’ = b³ (mod 4) . Determine the…

A:

Q: Let R be a relation defined on Z by xRy if and only if x-y=7k for kez. The equivalence class of [5]…

A: Given relation is xRy iff x-y=7k So, equivalence class of 5 will contain ....-9, -2, 5, 12, 19,....…

Q: Let R be an equivalence relation on A. Find the maximum number of classes R can make if JA| = 80 and…

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Q: Define a relation T on R as follows: for all x and y in R, x T y if and only if x2=y2. (a) Prove…

A: Given that a relation T on R such that for all x and y in R, xTy if and only if x2=y2 (a) We have to…

Q: Let a relation R on R2 be given by (x,y)R(x’,y’) provided that x-x’ and y-y’ are integers. Prove…

A:

Q: 18. Let R be the relation on R defined by xRy → x – y is an integer. Prove that R is an equivalence…

A:

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!